Sam: How can you state that my proof is invalid without showing where it breaks down?

Note that there is no radix point involved in my proof.

DecimalSeries(N) = 9/10 + 9/100 + 9/1000 + . . . + 9/10^N

HexSeries(N) = 15/16 + 15/256 + 15/4096 + . . . + 15/16^N

The above explicitly specify what recurring .99999 and recurring .FFFFF really represent, without using radix notation which is only a convenient shorthand notation.

For all values of N, it is obvious that DecimalSeries < HexSeries < 1

Hence there is a real number greater than DecimalSeries(N) and less than one, namely HexSeries(N).

Where does the above break down?

For what value of N does the above become invalid?

Perhaps you want to claim that HexSeries(N) = 1 for some value of N. Such a claim still leaves DecimalSeries(N) less that one.